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CHAPTER 3: Energy in bound system is quantized, light comes in quanta of energy / particles called photons

As far as I can see, our ideas are not in contradiction to the properties of the photoelectric effect observed by Mr. Lenard.

- Max Planck, 1905

Lenard got the Nobel prize that year for his work on cathode rays, but missed out on two of the most important discoveries in this kind of research (X-rays and the electron)

3.1 Discovery of the Electron (1897) 3.2 Determination of precise value of Electron Charge 3.3 Quantization of charge and mass Discovery of X Rays (1895), X-ray diffraction (1912)

and X-ray crystallography (1913) 3.4 Line Spectra 3.5 Blackbody Radiation 3.6 Photoelectric Effect 3.7 X-Ray Production and diffraction on crystals 3.8 Compton Effect

4.7: Frank-Hertz experiment, chapter 4

Pair production was already discussed in chapter 2 (special relativity)

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Modern Physics

waves or particles?

How come they are characteristic of atoms?

How come we can’t derive this from Maxwellian waves?

then applications, PH 312

the correct theory of matter at last

other loose ends in Classical Physics

J.J. Thompson (1897) electron Atoms have parts!

besides c = 1/srt(εμ) and minertial ≈ mdynamic

wave particle duality for massless particles

..

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Cathode Ray Experiments

In the 1890s scientists and engineers were familiar with “cathode rays”. These rays were generated from one of the metal plates in an evacuated tube across which a large electric potential had been established.

It was surmised that cathode rays had something to do with atoms. Different metals were used as cathode materials

It was known that cathode rays were deflected by magnetic and electric fields, could to some extent penetrate matter (but the tube itself required high vacuum for reproducible operations

Nobel prize 1905 Philip Eduard Anton von Lenard, who later became a Nazi, and would not find academic employment in the new Germany after WWII

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Apparatus of J.J. Thomson’s Cathode-Ray Experiment

Thomson used an evacuated cathode-ray tube to show that the cathode rays were negatively charged particles (electrons) by deflecting them in electric and magnetic fields.

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Thomson’s method of measuring the ratio of the electron’s charge to mass was to send electrons through a region containing a magnetic field perpendicular to an electric field.

J. J. Thomson’s Experiment

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An electron moving through the electric field is accelerated by a force:

Electron angle of deflection:

The magnetic field deflects the electron against the electric field force.

The magnetic field is adjusted until the net force is zero.

Charge to mass ratio:

Calculation of e/m

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Discovery of the Electrons

Electrons were discovered by J. J. Thomson 1897 Observed that cathode rays were

charged particles with a mass to charge ratio of about 1/2000 of the mass of an ionized hydrogen atom, i.e. a proton.

Same particle independent on what metal cathode it was extracted from, so atoms are not indivisible … should have an internal structure

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Millikan oil drop experiment

3.2: Determination of precise value of Electron Charge

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Calculation of the oil drop charge Used an electric field and

gravity to suspend a charged oil drop.

Mass is determined from Stokes’s relationship of the terminal velocity to the radius and density.

Magnitude of the charge on the oil drop.

Thousands of experiments showed that there is a basic quantized electron charge.

C

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Quantization of charge and mass

Current theories predict that charges are quantized in units (called quarks) of e/3, 2e/3 and e, but quarks are not directly observed experimentally. The charges of particles that have been directly observed are quantized in units of e.

The measured atomic weights are not continuous—they have only discrete values, which are close to integral multiples of a unit mass.

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Discovery of X Rays

X rays were discovered by Wilhelm Conrad Röntgen in 1895

Observed X rays emitted by cathode rays bombarding glass, found that they cannot be affected by electromagnetic fields, i.e. are not charged particles

Could not get diffraction effects, which would prove that they are Maxwellian waves, today we know this is due to their very small wavelength, very high frequency, very high energy

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Observation of X Rays

Wilhelm Conrad Röntgen studied the effects of cathode rays passing through various materials. He noticed that a phosphorescent screen near the tube glowed during some of these experiments. These rays were unaffected by magnetic fields and penetrated materials much more than cathode rays.

He called them x rays and deduced that they were produced by the cathode rays bombarding the glass walls of his vacuum tube.

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Röntgen’s X-Ray Tube

Röntgen constructed an x-ray tube by allowing cathode rays to impact the glass wall of the tube and produced x rays. He used xrays to image the bones of a hand on a phosphorescent screen.

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In the late afternoon of 8 November 1895, Röntgen carefully constructed a black cardboard. Before setting up the barium platinocyanide screen to test his idea, Röntgen darkened the room to test the opacity of his cardboard cover.

… he determined that the cover was light-tight and turned to prepare the next step of the experiment. It was at this point that Röntgen noticed a faint shimmering from a bench a few feet away from the tube. To be sure, he tried several more discharges and saw the same shimmering each time.

Striking a match, he discovered the shimmering had come from the location of the barium platinocyanide screen he had been intending to use next.

Nov./Dec 1895, he worked flat out, had a paper out by Christmas 1895,

Becquerel misunderstood the paper and discovered radioactivity in early 1896 as a result

First ever Nobel Prize in Physics, 1901

(1845–1923)

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Max Laue, Walter Friedrich, and Paul Knipping, 1912

With the benefit of Laue talking to Arnold Sommerfeld’s PhD student Peter Paul Ewald

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Nobel prizes

1914 to Max von Laue

1915 to Wilhelm Henry Bragg and Wilhelm Lawrence Bragg

http://iycr2014.org/

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Left: Hard sphere model of the diamond structure (space filling ≈ 34 %) by W. H. Bragg, Museum of the Royal Institution, London, photo by AndréAuthier. (Source: http://blog.oup.com/2013/08/100th-anniversary-first-crystal-structure-determinations-bragg/#sthash.50BE8wiT.6mYKaBKB.dpuf.) Right: Ball and stick model of the diamond structure. (Source: FIG. 7 in W. H. Bragg and W. L. Bragg, The structure of diamond, Proc. R. Soc. Lond. A 89, 277; published September 22, 1913.)

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Left: Reconstruction of the double helix model of deoxyribose nucleicacid containing some of the original metal plates; Right: used by Francis Crick and James Dewey Watson in 1953, (Source: Science Museum, http://www.sciencemuseum.org.uk/images/i045/10313925.aspx)

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This is actually a quasicrystal, you can tell from the 10 fold rotation symmetry

The image is a flat section through reciprocal/Fourier space

As a Fourier transform is a good mathematical model for the diffraction of X-rays by crystals (most ordinary condensed matter) and quasicrystals (some extraordinary condensed matter)

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3.3: Line Spectra (characteristic of atoms)

Chemical elements were observed to produce unique wavelengths of light when burned or excited in an electrical discharge.

Emitted light is passed through a diffraction grating with thousands of lines per ruling and diffracted according to its wavelength λ by the equation:

where d is the distance of line separation and n is an integer called the order number.

Note the significant difference of this formula to the Bragg equation

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If the material is very dilute there are discrete spectra: Ångström (~ 1860)

There is typically more emission lines than absorption lines, explanation later on when we understand atoms

dense material, i.e. hot “black body”, big- bang background radiation, sun if one does not look too carefully

For every absorption line, there is an emission line, but typically not the other way around

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1: this chapter, we need to learn more about the nature of light

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2: next chapter where we will use new insights on the nature of light

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Optical Spectrometer

Diffraction creates a line spectrum pattern of light bands and dark areas on the screen.

The line spectrum serves as a fingerprint of the gas that allows for unique identification of chemical elements and material composition.

gasses in tubes are typically quite dilute, discrete spectra

crystals are a different case, dhkl is spacing between atomic planes, easiest to understand in 3D reciprocal space

where d is “line spacing on the diffraction grating”, n and integer

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3.5: Blackbody Radiation

When matter is heated, it emits radiation.

A blackbody is a cavity in a material that only emits thermal radiation. Incoming radiation is absorbed in the cavity. Material is dense, so we expect a continuous spectrum

Blackbody radiation is interesting because the radiation properties of the blackbody are independent of the particular material! Physicists can study the distribution of intensity versus wavelength (or frequency) at fixed temperatures. Principle of a pyrometer to measure temperatures remotely.

emissivity є (є = 1 for idealized black body)

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Stefan-Boltzmann Law

Empirically, total power radiated (per unit area and unit wavelength = m-3) increases with temperature to power of 4: also intensity (funny symbol in Thornton-Rex)

This is known as the Stefan-Boltzmann law, with the constant σexperimentally measured to be 5.6705 × 10−8 W / (m2 · K4), Boltzmann derived the “form” of the empirical formula from classical physics statistics, it was not understood what the constant of proportionality actually meant

The emissivity є (є = 1 for an idealized black body) is simply the ratio of the emissive power of an object to that of an ideal blackbody and is always less than 1.

would be very interesting to know this function (intensity as a function of and T)

Watt per m2

power = current times voltage, time derivative of mechanical work, unit of watt

Energy density u(T) = 4/c ε σ T4 in Ws m-3

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Wien’s Displacement Law

The intensity (λ, T) is the total power radiated per unit

area per unit wavelength at a given temperature.

Wien’s displacement law: The maximum of the distribution shifts to smaller wavelengths as the temperature is increased.

Most interesting, what is the mathematical function that describes all of these curves??

the higher T, the shorter , no classical physics justification provided

Wm-2

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So that must be approximately the black body radiation from the sun.

Why are our eyes particularly sensitive to green light of 550 nm? Because life evolved on earth receiving radiation from the sun that peaks at this particular wavelength.

Unit of intensity Watt per area

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Wilhelm Wien, Physics Nobel prize 1911

Problem: two fitting constants for which people struggled to find physical meaning, only Max Planck in 1900 provided some kind of an explanation of ß in the limit of very short wavelengths, to be revisited later

Wien’s black body radiation formula with two free fitting parameters, to be determined by experiment

Watt-second (energy)Per m3 after multiplication of intensity (Wm-2) with 4/c, (factor 4 from steradiant = 4π sr for surface of a sphere)

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A continuous spectrum of electromagnetic radiation, for the very low temperature of the universe (approx. 3 K) far away from stars

Leaders of COBE project, George Smootand John Mather, received 2006 Physics Nobel prize, now even better measurements

http://en.wikipedia.org/wiki/Cosmic_Background_Explorer

Very slight anisotropy, i.e. slightly different blackbody radiation temperature

Cosmic Background Explorer(COBE)

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Rayleigh-Jeans Formula Lord Rayleigh (John Strutt) and James Jeans used the classical

theories of electromagnetism and thermodynamics to show that theblackbody spectral distribution should be, 1905

It approaches the data at longer wavelengths, but it deviates very badly at short wavelengths. This problem for small wavelengths became known as “the ultraviolet catastrophe” and was one of the outstanding problems that classical physics could not explain around 1900s, implication: light may not be a Maxwellian wave

With k as Boltzmann constant:

1.38065 10-23 J/K

kT at room temperature (293 K) ≈ 25 meV

http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html

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Planck had the very best experimental data (over the widest available wavelength range available, he started out with Boltzmann’s statistical methods, but that didn’t work. By a curve fit over the best available data, he obtained the following formula that describes the totality of the blackbody radiation data very well, to provide a theoretical explanation, he assumed that the radiation in the cavity was emitted (and absorbed)

by some sort of “oscillators” that were contained in the walls.

This formula/result flies in the face of classical physics because:

1) The oscillators (of electromagnetic origin) can only have certain discrete energies determined by En = nhf, where n is an integer, fis the frequency, and h is now called Planck’s constant. h = 6.6261 × 10−34 J·s (Ws2)

2) The oscillators can absorb or emit energy only in discrete multiples of the quanta of energy given by

Max Planck’s Radiation Law

Planck’s radiation law

and a Nobel Prize for M. Planck, 1918 a hint on quantized energy in atoms

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rad

How does this sound to a classical physicist?

Transitions in energy of a harmonic quantum oscillator are just E = hf, this assumption will be confirmed by standard 3D Quantum Mechanics at the end of this course. Planck didn't know about zero point energy and Heisenberg’s uncertainty principle, so his zero level is off.

Approx. simple harmonic motion for small swings, g is a “constant” of the bob Earth (bound) system, mechanical energy can have any value

1)exp(

8),(

4

Tkhfhf

Tfu

B

1)exp(

8),(

5

Tkhchc

Tu

B

1)exp(

8),(

4

thermal

photons

photons

EE

ETfu

Versions of Planck’s black body radiation law

unit of energy/volume Conversion factor = c/4

4/c u(λ,T)

Watt per area

In Serway et al. and other books u(f,T) is defined differently, but also correctly

1

2

3

}1){exp(2

)( Bk

h

c

hTB

d

dTBTB )()(

1

5

2

}1){exp(2

)( Tk

hchcTB

B

Surface brightness per unit frequency (Greek symbol Ny) or wavelength (λ) with respect to the projected area of the surface, again a conversion factor 4/c can be applied – as in next slide

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Assumes light is a only a wave phenomenon

is frequency, also f in many books

Somewhat similar to Wien’s empirical formula, short wavelengths

Maximum is found from derivative of the Plank law formula, used in pyrometers

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Energy density as function of frequency and temperature

exp(x) ≈ 1 + x for very small x, i.e. when h → 0, d. h.

classical physics (also for f very small for any T)

Small frequency means large wavelength, behave more like a wave, high frequency means it does not behave like a wave (it needed an Einstein to figure out what is going on)

h = 4.135667 10-15 eV so rather small but not zero, we do not live in a classic physics world although that is all we can comprehend directly

Total energy for a photon: E = h f, but it takes a Genius (A. Einstein) to realize that. Everybody else cannot make sense of Planck’s assumption of quantization of the energy levels of his “resonators” at that time. Note that Rayleigh-Jeans formula is given correctly only in 1905

Rayleigh-Jeans formula from classical physics, for hf

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if f is very large, wavelength very short (for any kind of T), the exponential is really large, so that the -1 term might be neglected to a good approximation

Wien’s empirical law with two fitting parameters somewhat similar results, for very energetic photons, which behave much like classical particles, classical physics, for hf >> kT

So the Planck formula get both regions, short wavelengths (high frequencies) and large wavelengths (small frequencies) right !!

Einstein saw the deeper implications of the peculiar form of the Energy density (or Intensity per area) versus wavelength curve first in 1905

Removing the -1 changes the distribution function for bosons to classical particles

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N.B. there are only a very few fundamental constants, sure Stefan’s constant was not one of them, kB , c, and h are, loosely speaking these three constants are about the thermal behavior of particles, travel of

light waves, and “strength” of the wave – particle coupling / duality

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1)exp(

8),(

5

Tkhchc

Tu

B

Einstein realized that Planck’s formula describes something never before “seen” in physics, at very small wavelengths, we have “particle like” behavior, at very large wavelengths, we have “wave like” behavior

So light shows wave – particle duality properties, but is neither a classical wave nor a classical particle, it’s a quantum mechanical entity,

Light is a stream of “mass less” particles, which are allowed by special relativity, that each individually possess total energy E = h f, which is all kinetic, rushing around with the speed of light for any observer (in an inertial frame of reference, with general relativity in any frame of reference)! Isn’t that exiting?? Accelerator equation for mass less particles E = pc, with

c = f, we get p = h/ as “hallmark” of wave-particle duality

How come that photons do not have rest-mass? Their total energy is E = hf, with c = λf

220

2020 )()(1 cvmcmc

m

For particles with rest mass Because v < c, there is always a positive value, for rest-mass for relativistic particles other than light photons

For mass-less particles

0)()(1)()(1 2222222

2?

0 cc

hfhf

cc

hhf

cmlight

Because λ = c/f

We have two new equations for the total energy (which is all kinetic) and the momentum of a photon from Einstein’s realization that light particles jump around at the speed of light in discrete bundles with well defined energy and momentum, but no mass

Mass is not a continuum either, it’s a bit like coins, 1¢, 2¢, 5¢, 10¢, 25¢ coins, but not 31¢ coins (but there is also binding energy when elemental particles stick together)

Charge of the (free) elemental particles, electrons, protons, positrons, … come in units of the elemental charge. So they are quantized in integers (oil drop experiment).

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http://usatoday30.usatoday.com/tech/science/genetics/2008-05-08-platypus-genetic-map_N.htm

“Australia's unique duck-billed platypus is part bird, part reptile and part mammal according to its gene map.

The platypus is classed as a mammal because it has fur and feeds its young with milk, but they hatch from eggs. It flaps a beaver-like tail. But it also has bird and reptile features — a duck-like bill and webbed feet, and lives mostly underwater. Males have venom-filled spurs on their heels.”

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3.6: Photoelectric Effect

Methods of electron emission:

Thermionic emission: Application of heat allows electrons to gain enough energy to escape.

Secondary emission: The electron gains enough energy by transferfrom another high-speed particle that strikes the material from outside.

Field emission: A strong external electric field pulls the electron out of the material.

Photoelectric effect: Incident light (electromagnetic radiation) shining on the material transfers energy to the electrons, allowing them to escape.

Electromagnetic radiation interacts with electrons within metals and gives the electrons increased kinetic energy. Light can give electrons enough extra kinetic energy to allow them to escape. We call the ejected electrons photoelectrons.

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Experimental Setup

variable voltage,polarity such that negative charge is applied to the collector, stopping the electrons = stopping (retarding) potential

How much energy is needed to get electrons out of a metal

http://www.walter-fendt.de/ph14e/photoeffect.htm

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Experimental Results

1) Only sufficiently energetic light makes an effect, there can be an enormous intensity of light that is not sufficiently energetic and no effect is observed

2) The kinetic energies of the photoelectrons are independent of the light intensity.

3) The maximum kinetic energy of the photoelectrons, for a given emitting material, depends only on the frequency of the light.

4) The smaller the work function φ of the emitter material, the smaller is the threshold frequency of the light that can eject photoelectrons.

5) When the photoelectrons are produced, however, their number is proportional to the intensity of light (electronic light intensity measurement are based on that).

6) The photoelectrons are emitted almost instantly following illumination of the photocathode, independent of the intensity of the light.

http://lectureonline.cl.msu.edu/~mmp/kap28/PhotoEffect/photo.htm

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Same emitter material (V0), same kind of light (f or ) with 1, 2, and 3 fold intensity, photocurrent intensity depends on incoming light intensity if sufficiently energetic light hf has been used

Three different emitter materials, three different kinds of light (f) are required to get a photocurrent for the same intensity of the incoming light

Experimental Results

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Experimental Results

Different materials have different work functions, i.e. amounts of energy required to allow an electron to escape a metal block, note that the slope of all of these curves is just h

Analogous to first graph from pervious slide (which showed just three different values of intensities of the incoming light, Photocurrent is linear function of light intensity if it is sufficiently energetic to overcome the work function (stopping potential)

Submitted on 13 Sep 2019High Precision Determination of the Planck Constant by Modern

Photoemission Spectroscopy

Jianwei Huang, Dingsong Wu, Yongqing Cai, Yu Xu, Cong Li, Qiang Gao, Lin Zhao, Guodong Liu, Zuyan Xu, X. J. Zhou

The Planck constant, with its mathematical symbol h, is a fundamental constant in quantum mechanics that is associated with the quantization of light and matter. It is also of fundamental importance to metrology, such as

the definition of ohm and volt, and the latest definition of kilogram. …Through the direct use of the Einstein's photoelectric equation, the Planck constant is determined by measuring accurately … using light sources with

various photon wavelengths. The precision of the measured Planckconstant, 6.62610(13)×10−34 J⋅s, is four to five orders of magnitude improved from the previous photoelectric effect measurements…

https://arxiv.org/abs/1909.06286

In 2020, the numerical value of the Planck constant has been fixed, with finite significant figures, just like the speed of light and the second before.

The kilogram has also been redefined. Under the present technical definition of the kilogram: one kilogram is now defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit J⋅s, and the equivalency of E = mc2 and E = hf for photons into account. A certain amount of photons are equivalent to rest mass of 1 kg. Future refinements will be in the number of photons and Avogadro’s number (and the chemical mole) as h and c are now fixed.

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Attempt of a classical interpretation The existence of a threshold frequency is completely inexplicable in

classical theory.

Classical theory predicts that the total amount of energy in a light wave increases as the light intensity increases.

Classical theory would predict that for extremely low light intensities, a long time would elapse before any one electron could obtain sufficient energy to escape. We observe, however, that the photoelectrons are ejected almost immediately.

Conclusion, light can not be a Maxwellian wave, must be something else, a stream of particles (as though through by Newton in deriving geometric (ray-) optic

On the other hand, light shows typical wave properties, i.e.

interference, so what light seems to be depends on the experiment?

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Einstein’s Interpretation

Einstein suggested that the electromagnetic radiation field is quantized into particles called photons. Each photon possesses the quantum of total energy (which is all kinetic as they do not have mass):

where f is the frequency of the light and h is Planck’s constant.

The photon travels at the speed of light in a vacuum, defining its wavelength and frequency

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Conservation of total energy yields:

where is the work function of the metal.

Explicitly the energy is (as v is small, no special relativity treatment)

The retarding (stopping) potentials measured in the photoelectric effect are the opposing potentials needed to stop the most energetic electrons.

Einstein’s analysis

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Light Quantum Theory Interpretation

The kinetic energy of the electron does not depend on the light intensity at all, but only on the light frequency and the work function of the material.

Einstein in 1905 predicted that the stopping potential was linearly proportional to the light frequency, with a slope h, the same constant found by Planck.

From this, Einstein concluded that light consists of particles with energy:

a Nobel Prize for A. Einstein, 1921 (none for his other great achievements as people had difficulty with special/general relativity and its experimental verification for a long time)

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Millikan himself on a more informal occasion: ”I spent ten years of my life testing that 1905 equation of Einstein’s and contrary to all my expectations, I was compelled in 1915 to assert its unambiguous verification in spite of its unreasonableness since it seemed to violate everything we knew about the interference of light.

When Millikan’s Nobel Prize came to pass (1923), his Nobel address contained passages that showed his continuing struggle with the meaning of his own achievement: “This work resulted, contrary to my own expectation, in the first direct experimental proof…of the Einstein equation and the first direct photo-electric determination of Planck’s h.”(Nobel prize for precision measurements of both elemental charge and h)

"All this nonsense makes no sense!" Millikan_Ariel in The Blegdamsvej Faust

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3.7: X-Ray Production An energetic electron passing through matter will radiate photons and lose kinetic

energy which is called bremsstrahlung, from the German word for “braking radiation.” Linear momentum must be conserved, but a nucleus of an atom absorbs very little energy, and it is ignored. The final energy of the electron is determined from the conservation of energy to be

An electron that loses a large amount of energy will produce an X-ray photon of very short wavelength. Current passing through a filament produces copious numbers of electrons by thermionic emission. These electrons are focused by the cathode structure into a beam and are accelerated by potential differences of tens of thousands of volts until they impinge on a metal anode surface, producing x-rays by bremsstrahlung as they are deflected in the anode material.

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“Inverse Photoelectric Effect” Conservation of energy requires that the

electron kinetic energy equal the maximum photon energy where we neglect the work function because it is very small. This yields the Duane-Hunt limit which was first found experimentally. The photon wavelength depends only on the accelerating voltage V0 and is the same for all targets.

The really interesting bits are the intense and narrow “X-ray lines”, which are characteristic of the anode material and are called characteristic X-rays, one can use them either to identify the material in an electron microscope or use them for quasi-monochromatic X-ray diffraction experiments

Mo

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W Mo

62Any good theory of matter (atoms) needs to explain this peculiar behavior !! next chapter

Because of its high intensity, it makes sense to use only Kα X-rays (nearly monochromatic) for X-ray crystallography

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Apparent gaps in the periodic table of elements

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3.8: Compton Effect

as we have seen from Millikan’s statements, the physics establishment was baffled by Einstein’s wave-particle duality ideas for light, i.e. light itself is quantized, not only the energy in a harmonic oscillator as needed by Planck in order to derive the right kind of formula for black-body radiation (where he obtained a first value of h from a simple curve fit)

The photoelectric effect would only give us the maximal kinetic energy of the photoelectrons, a more direct proof that light is pure kinetic energy of certain sizes was needed !!!

i.e. a direct collision experiment of sufficiently energetic light particles (best with very high intensity to simplify the detection) with some small particle that possesses mass, so let’s take characteristic X-rays and electrons

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Classical physics prediction: loosely speaking: the E vector would pick the electron up, shake it a bit and a new wave of longer wavelengths would be created that radiates outwards in a spherical manner, i.e. no directional difference whatsoever

If light has also particle properties (in addition to being a Maxwellian wave), a photon should collide with the electron, total energy and momentum needs to be conserved, special relativity needs to be involved as the “light particle” moves with the speed of light

1

2

We cannot keep track of the electron, so no way to measure Ф, but we can do spectroscopy on the inelastically scattered X-rays as a function of angle θ

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as their binding energy, a few eV

is much less than the energy of the incoming photon

according to special relativity

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New left side only as right side is simple, then cancelling E0 dividing by c2 and resolving for pe

2 , whereby

2

00212

212

2021

2021

)(2)(

})({)(

EEppcppc

EppcEcpcp

2

221

2

12

21 2)( pppppp Note that we have two independent equations for pe

2 , by using them both, we get rid of p12 and p2

2

and gain an interesting cos θ term, this also clarifies that the result will be independent on the initial wavelength of the light, we will just derive a change in wavelengths that is “hiding” in the products of p1p2

and

accelerator equation of special relativity

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20 mcE 2

21

21 )cos1()(

mc

hc

pp

pph

khp That is now just for photons, later on we will see that it is also valid for particles, it’s then the De Broglie equation, next but one chapter

122121

}11{

mc

hhhh )cos1(21

21

2

2

1

2

Just what we need for the left side (the right side is already OK), to finally obtain

2 = ’ > 1 so positive as it must

h/mc is a derived constant, Compton wavelength for a certain type of particle with (rest) mass m, what’s interesting is the shift, the larger the mass, the smaller the shift, and 800 pound gorilla is not knocked over by a single photon

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The calcite crystal picks out a certain narrow wavelength range only, the relative shift of the inelastically scattered waves for a variation of Φ is what the experiments wants to demonstrate, (note as a aside that the same book called this angle earlier θ)

Diffraction spots from R are undesirable, so detector gets shielded from them

Scattering is strongly forward, but with lower and lower intensity in all directions (except for the Bragg peaks)

The inelastic scattering has always two components, from the tightly bound atoms and the lightly bound electrons, the higher Ф, the easier these components are taken apart

Note that we treat the X-ray photons as waves

Here the X-ray photons are particles

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In (a) only one peak, transmitted X-rays plus inelastically scattered X-rays with have a slightly longer wavelength from both atoms and electrons overlap

In (d) finally, the inelastically scattered X-rays from the atoms and electrons clearly separate the spacing's between these two peaks is in good agreement with

whereby θ is most definitely not the Bragg angle as the spectroscopy principle of the analysis has nothing to do with the diffraction effect within the experiment, was earlier called Фin the derivation

intensities drop significantly as variable angle Фincreases, but this is just an experimental difficulty

Not on the same abso-lute scale

and is very close to that of the characteristic X-ray source

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Why were X-rays required for Compton’s scattering experiments?

Actually also for ≈ 0.0712 nm for Mo Kαused in Compton experiments about 100 years ago, highest respect to overcoming the experimental difficulties !

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Nobel prize 1927 for Arthur Holly Compton, so there is no doubt that the photons (mass less particles compliant with special relativity) are indeed behaving like particles in collisions

as Einstein had predicted already in 1906: “If a bundle of radiation causes a molecule to emit or absorb an energy packet hf, then momentum of quantity hf/c is transferred to the molecule, directed along the line of motion of the bundle for absorption and opposite to the bundle for emission”

(for photons, p = E/c from special relativity 1905)

As people had enormous difficulties with this, the great master (A. E.) wrote in 1911: “I insist on the provisional nature of this concept which does not seem reconcilable with the experimentally verified consequences of the wave theory.”

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We have seen in this chapter irrefutable evidence for Einstein’s wave-particle duality nature of light and its consistency with special relativity

we dealt with the latter in the previous chapter

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4.7: Frank-Hertz experiment, 1914

Nobel prize in 1925

Again something peculiar hinting about quantization of energy in bound systems, (i.e. inside Hg atoms), again probed by collisions between particles, this time cathode rays (i.e. accelerated electrons) and a dilute gas of Hg, also confirming Einstein’s E = hf equation and with it the existence of photons, (i.e. wave particle duality), just one year after Bohr’s three paper sequence on the structure of the hydrogen atom

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4.7: Atomic Excitation by Electrons

Some electrons pass through the grid and reach the collector, the purpose of the voltage between the grid and collector is similar to the stopping potential in the photoelectric effect: to control kinetic energy of the electrons (but now some electrons get through as any grid has holes.

V

Measurement signal is a current in an electrometer in dependence of a variable acceleration voltage

First with glass tube, was replaced by quartz glass tubing, which allows ultraviolet light to be measured outside the tube, sure ultraviolet light of

might be different, just small

was detected above 4.9 eV (V0) and higher. So that light’s energy must correspond to an internal transition in the Hg, so atoms should have quantized energy levels !!

hc useful constant combination: 1239.8 eV

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As acceleration voltage increases from 0 to just below approx. 4.9 eV (+1.5 eV from the collector/grid, also modifications due to work function of filament and collector), the registered current increases, more electrons get through

at (and above) 4.9 eV (+1.5), electrons loose essential all of their kinetic energy to the much heavier Hg atoms, the registered current drops to nearly zero, some electrons pass by Hg atoms without collision

when acceleration voltage is increased beyond 4.9 eV (+1.5 eV), the registered current increases again until it drops at two times 4.9 eV (+1.5 eV), the spacing between subsequent peaks is approx. 4.9 eV (regardless of the small work function of the collector and other corrections),

One could estimate the density of Hg atoms in the dilute gas from this

compare with sketch of experimental data

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Excitation of Hg atoms by Electrons Explanation, only electrons with sufficient kinetic energy 4.9 eV

loose it to Hg atoms, exited Hg atoms just take in that quantum of energy (in the low voltage range), after a short time they radiate it off again, the radiation intensity is zero before kinetic energy 4.9 eV is reached and exceeded.

What’s important about this kind of plot is that the spacing between the maxima is just 4.9 eV (not were they actually occur, because the latter is a function of the slightly negative potential between collector and grid)

There are multiple peaks in such a plot because a particular electron might loose 4.9 eV to a Hg atom, be accelerated again, loose again those 4.9 eV to another Hg atom, …. Do realize that Hg “takes in only 4.9 eV, no less no more”and looses 4.9 eV no less no more in radiation – how come: that must correspond to some internal transition on the basis of some internal structure which any good theory of an atom must be able to explain, next chapter

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Now this “kind of effect” is used to characterize elements and compounds, e.g. in an analytical transmission electron microscope

http://en.wikipedia.org/wiki/Electron_energy_loss_spectroscopy

Doing spectroscopy on the emitted light is called cathodoluminescents, i.e. another modern characterization technique in SEM where you do not have many transmitted electrons

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Anybody in this class who has not heard about the Bohr model of the hydrogen atom before?????

My class is about how radical that model was for its time and where its limitations are, modern physics is not separated in chapters, it all must fit together, that’s how these people have progressed in the past and that’s the only way to make more progress in the future, also about essence of models